Topic
Piecewise
About: Piecewise is a research topic. Over the lifetime, 21064 publications have been published within this topic receiving 432096 citations. The topic is also known as: piecewise-defined function & hybrid function.
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TL;DR: In this paper, the authors studied piecewise deterministic Markov processes with state space R x E where E is a finite set and the continuous component evolves according to a smooth vector field that is switched at the jump times of the discrete coordinate.
Abstract: We study a class of Piecewise Deterministic Markov Processes with state space Rd x E where E is a finite set. The continuous component evolves according to a smooth vector field that is switched at the jump times of the discrete coordinate. The jump rates may depend on the whole position of the process. Working under the general assumption that the process stays in a compact set, we detail a possible construction of the process and characterize its support, in terms of the solutions set of a differential inclusion. We establish results on the long time behaviour of the process, in relation to a certain set of accessible points, which is shown to be strongly linked to the support of invariant measures. Under Hormander-type bracket conditions, we prove that there exists a unique invariant measure and that the processes converges to equilibrium in total variation. Finally we give examples where the bracket condition does not hold, and where there may be one or many invariant measures, depending on the jump rates between the flows.
164 citations
TL;DR: A comprehensive derivation is presented of normal form maps for grazing bifurcations in piecewise smooth models of physical processes, which links grazings with border-collisions in nonsmooth maps.
Abstract: A comprehensive derivation is presented of normal form maps for grazing bifurcations in piecewise smooth models of physical processes. This links grazings with border-collisions in nonsmooth maps. Contrary to previous literature, piecewise linear maps correspond only to nonsmooth discontinuity boundaries. All other maps have either square-root or $(3/2)$-type singularities.
164 citations
TL;DR: In this article, a special displacement with bounded deformation is approximated with a sequence (un)n⩾1 of piecewise continuous displacements whose jump sets Jun are (relatively) closed, with un and e(un) converging strongly in L2, respectively to u and e.
164 citations
TL;DR: For a model convection-dominated singularly perturbed convectiondiffusion prob- lem, it is shown in this article that crosswind smear in the numerical streamline diffusion finite element method is minimized by introducing a judicious amount of artificial crosswind diffusion.
Abstract: For a model convection-dominated singularly perturbed convection-diffusion prob- lem, it is shown that crosswind smear in the numerical streamline diffusion finite element method is minimized by introducing a judicious amount of artificial crosswind diffusion. The ensuing method with piecewise linear elements converges with a pointwise accuracy of almost h 5/4 under local smoothness assumptions. 1. Introduction. The streamline diffusion method is a finite element method for convection-dominated convection-diffusion problems which combines formal high accuracy with decent stability properties. The method was introduced in the case of stationary problems by Hughes and Brooks (7), cf. Raithby and Torrance (14) and Wahlbin (15) for earlier thoughts in this direction. The mathematical analysis of the method was started in Johnson and Navert (8) and continued with extensions to, e.g., time-dependent problems in Navert (12), Johnson, Navert and Pitkaranta (9) and Johnson and Saranen (10). In these papers local error estimates in L2 of order O(h k 1/2), in regions of smoothness, with piecewise polynomial finite elements of degree k, were derived, together with estimates stating, as a typical example, that in the zero diffusion limit a sharp discontinuity in the exact solution across a streamline will be captured in a numerical crosswind layer of width 0(h1/2), essentially. The purpose of the present paper is first to improve the result just mentioned on numerical crosswind smear to 0(h3/4). The improvement from 0(h1/2) to 0(h3/4) is obtained by adding a small amount, 0(h3/2), of artificial crosswind diffusion to the method. In the piecewise linear case (k = 1) this does not destroy the known O( h3/2) accuracy in L2 in smooth regions. Using our first result, we then obtain our second main result, localized pointwise error estimates of order 0(h5/4) in regions of smoothness. (The previously known best pointwise error estimate in the piecewise linear situation is 0(h1/2).) Another consequence is a global L,-estimate of order 0(h'/2) in the presence of typical crosswind and downwind singularities. We shall consider the model problem of finding u = u(x, y) such that
164 citations
TL;DR: This paper presents a recurrent neural network for solving strict convex quadratic programming problems and related linear piecewise equations with a one-layer structure with a low model complexity and shown to have a finite-time convergence and exponential convergence.
163 citations